Hidden nonlinear su(2|2) superunitary symmetry of N=2 superextended 1D Dirac delta potential problem

TL;DR

This paper reveals a hidden nonlinear su(2|2) superunitary symmetry in a simple N=2 superextended 1D Dirac delta potential system, characterized by three different Z2-gradings and multiple sets of integrals of motion.

Contribution

It uncovers a novel hidden nonlinear su(2|2) superunitary symmetry with multiple gradings in a basic supersymmetric quantum system, expanding understanding of hidden symmetries.

Findings

The system admits three different Z2-gradings.

It features three sets of 8 bosonic and 8 fermionic integrals of motion.

The nonlinear superalgebra reduces to 2D Euclidean superextended Poincaré algebra on the ground state.

Abstract

We show that the N=2 superextended 1D quantum Dirac delta potential problem is characterized by the hidden nonlinear $su(2|2)$ superunitary symmetry. The unexpected feature of this simple supersymmetric system is that it admits three different $\mathbb Z_2$-gradings, which produce a separation of 16 integrals of motion into three different sets of 8 bosonic and 8 fermionic operators. These three different graded sets of integrals generate two different nonlinear, deformed forms of $su(2|2)$, in which the Hamiltonian plays a role of a multiplicative central charge. On the ground state, the nonlinear superalgebra is reduced to the two distinct 2D Euclidean analogs of a superextended Poincar\'e algebra used earlier in the literature for investigation of spontaneous supersymmetry breaking. We indicate that the observed exotic supersymmetric structure with three different $\mathbb Z_2$-gradings can be useful for the search of hidden symmetries in some other quantum systems, in particular, related to the Lam\'e equation.